jmc

We can write the equation as $4x^3-41x^2+10x-1989=0.$ Then by the Integer Root Theorem, the integer root must be a factor of $1989=3^2\cdot 13\cdot 17.$

Also, we can factor both sides as $$(x-10)(x)(4x-1)=3\cdot 3\cdot 13\cdot 17.$$in particular, $x-10$ must be a positive factor of the right-hand side. Trying $x-10=3$ gives us $x=\boxed{13},$ which works.